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The purpose of this study is to extend the concept of a generalized Lie $3-$algebra, known to the divisional algebra of the octonions $\mathbb{O}$, tosplit-octonions $\mathbb{SO}$, which is non-divisional. This is achievedthrough the unification of the product of both of the algebras in a singleoperation. Accordingly, a notational device is introduced to unify the productof both algebras. We verify that $\mathbb{SO}$ is a Malcev algebra and werecalculate known relations for the structure constants in terms of theintroduced structure tensor. Finally we construct the manifestlysuper-symmetric $\mathcal{N}=1\;\mathbb{SO}$ affine super-algebra. Anapplication of the split Lie $3-$algebra for a Bagger and Lambert gauge theoryis also discussed.

Lie $3-$algebra and super-affinization of split-octonions